The Eight Cayley–Dickson Doubling Products

The purpose of this paper is to identify all eight of the basic Cayley–Dickson doubling products. A Cayley–Dickson algebra \({\mathbb{A}_{N+1}}\) of dimension \({2^{N+1}}\) consists of all ordered pairs of elements of a Cayley–Dickson algebra \({\mathbb{A}_{N}}\) of dimension \({2^N}\) where the product \({(a, b)(c, d)}\) of elements of \({\mathbb{A}_{N+1}}\) is defined in terms of a pair of second degree binomials \({(f(a, b, c, d), g(a, b, c,d))}\) satisfying certain properties. The polynomial pair\({(f, g)}\) is called a ‘doubling product.’ While \({\mathbb{A}_{0}}\) may denote any ring, here it is taken to be the set \({\mathbb{R}}\) of real numbers. The binomials \({f}\) and \({g}\) should be devised such that \({\mathbb{A}_{1} = \mathbb{C}}\) the complex numbers, \({\mathbb{A}_{2} = \mathbb{H}}\) the quaternions, and \({\mathbb{A}_{3} = \mathbb{O}}\) the octonions. Historically, various researchers have used different yet equivalent doubling products.     

Factorizations of the Complete Graph into C 5-Factors and 1-Factors

In this paper, we show that K_10n can be factored into C₅-factors and 1-factors for all non-negative integers and satisfying 2+=10n–1.Research partially supported by an NSF-AWM Mentoring Travel Grant     

On some algebraic properties of CM-types of CM-fields and their reflexes

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The purpose of this paper is to show that the reflex fields of a given CM-field K are equipped with a certain combinatorial structure that has not been exploited yet.The first theorem is on the abelian extension generated by the moduli and the b-torsion points of abelian varieties of CM-type, for any natural number b. It is a generalization of the result by Wei on the abelian extension obtained by the moduli and all the torsion points. The second theorem gives a character identity of the Artin L-function of a CM-field K and the reflex fields of K. The character identity pointed out by Shimura (1977) in [10] follows from this.The third theorem states that some Pfister form is isomorphic to the orthogonal sum of defined on the reflex fields _⊕Φ∈ΛK^∗(Φ). This result suggests that the theory of complex multiplication on abelian varieties has a relationship with the multiplicative forms in higher dimension.

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For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=IIwksVYV5YE.     

A Hanf number for saturation and omission: the superstable case

Suppose is a triple of two theories in vocabularies with cardinality λ, and a τ₁‐type p over the empty set that is consistent with T₁. We consider the Hanf number for the property “there is a model M₁ of T₁ which omits p, but is saturated”. In [2], we showed that this Hanf number is essentially equal to the Löwenheim number of second order logic. In this paper, we show that if T is superstable, then the Hanf number is less than .     

On the zeta function associated with module classes of a number field

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The goal of this note is to generalize a formula of Datskovsky and Wright on the zeta function associated with integral binary cubic forms. We show that for a fixed number field K of degree d, the zeta function associated with decomposable forms belonging to K in d−1 variables can be factored into a product of Riemann and Dedekind zeta functions in a similar fashion. We establish a one-to-one correspondence between the pure module classes of rank d−1 of K and the integral ideals of width <d−1. This reduces the problem to counting integral ideals of a special type, which can be solved using a tailored Moebius inversion argument. As a by-product, we obtain a characterization of the conductor ideals for orders of number fields.

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For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=RePyaF8vDnE.     

Extending the Erdős–Ko–Rado theorem

Let be a k‐uniform hypergraph on n vertices. Suppose that holds for all . We prove that the size of is at most if satisfies and n is sufficiently large. © 2005 Wiley Periodicals, Inc. J Combin Designs     

On Li's criterion for the Riemann hypothesis for the Selberg class

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In this paper, we shall prove a generalization of Li's positivity criterion for the Riemann hypothesis for the extended Selberg class with an Euler sum. We shall also obtain two arithmetic expressions for Li's constants , where the sum is taken over all non-trivial zeros of the function F and the indicates that the sum is taken in the sense of the limit as T→∞ of the sum over ρ with |Imρ|?T. The first expression of λF(n), for functions in the extended Selberg class, having an Euler sum is given terms of analogues of Stieltjes constants (up to some gamma factors). The second expression, for functions in the Selberg class, non-vanishing on the line , is given in terms of a certain limit of the sum over primes.

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For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=EwDtXrkuwxA.     

On Weighted Average Interpolation with Cardinal Splines

Given a sequence of data \(\{ y_{n} \} _{n \in \mathbb{Z}}\) with polynomial growth and an odd number \(d\), Schoenberg proved that there exists a unique cardinal spline \(f\) of degree \(d\) with polynomial growth such that \(f ( n ) =y_{n}\) for all \(n\in \mathbb{Z}\). In this work, we show that this result also holds if we consider weighted average data \(f\ast h ( n ) =y_{n}\), whenever the average function \(h\) satisfies some light conditions. In particular, the interpolation result is valid if we consider cell-average data \(\int_{n-a}^{n+a}f ( x ) dx=y_{n}\) with \(0< a\leq 1/2\). The case of even degree \(d\) is also studied.     

New gap results on the 4‐dimensional sphere

A result showed by M. Gursky in 4 ensures that any metric g on the 4‐dimensional sphere satisfying and is isometric to the round metric. In this note, we prove that there exists a universal number i ₀ such that any metric g on the 4‐dimensional sphere satisfying and is isometric to the round metric. Moreover, there exists a universal such that any metric g on the 4‐dimensional sphere with nonnegative sectional curvature, and is isometric to the round metric. This last result slightly improves a rigidity theorem also proved in 4 .     

Strengths and Weaknesses of LH Arithmetic

In this paper we provide a new arithmetic characterization of the levels of the og‐time hierarchy (LH). We define arithmetic classes and that correspond to ‐LOGTIME and ‐LOGTIME, respectively. We break and into natural hierarchies of subclasses and . We then define bounded arithmetic deduction systems ′ whose ‐definable functions are precisely B( ‐LOGTIME). We show these theories are quite strong in that (1) LIOpen proves for any fixed m that , (2) TAC, a theory that is slightly stronger than ′ whose (LH)‐definable functions are LH, proves LH is not equal to ‐TIME(s) for any m> 0, where 2^s ∈L, s(n) ∈ ω(log n), and (3) TAC proves LH ≠ for all k and m. We then show that the theory TAC cannot prove the collapse of the polynomial hierarchy. Thus any such proof, if it exists, must be argued in a stronger systems than ours.     

Two Questions of Erdős on Hypergraphs above the Turán Threshold

For ordinary graphs it is known that any graph G with more edges than the Turán number of must contain several copies of , and a copy of , the complete graph on vertices with one missing edge. Erd?s asked if the same result is true for , the complete 3‐uniform hypergraph on s vertices. In this note, we show that for small values of n, the number of vertices in G, the answer is negative for . For the second property, that of containing a , we show that for the answer is negative for all large n as well, by proving that the Turán density of is greater than that of .     

Compactness Conditions for the Green Operator in Lp(ℝ) Corresponding to a General Sturm–Liouville Operator

We consider the equation ℝ, where , for ℝ, (ℝ), (ℝ), (ℝ), (ℝ) := C(ℝ)). We give necessary and sufficient conditions under which, regardless of , the following statements hold simultaneously: I) For any (ℝ) Equation (0.1) has a unique solution (ℝ) where $\int ^{\infty}_{-\infty}$ ℝ. II) The operator (ℝ) → (ℝ) is compact. Here is the Green function corresponding to (0.1). This result is applied to study some properties of the spectrum of the Sturm–Liouville operator.     

The quasi‐randomness of hypergraph cut properties

Let satisfy and suppose a k‐uniform hypergraph on n vertices satisfies the following property; in any partition of its vertices into k sets of sizes , the number of edges intersecting is (asymptotically) the number one would expect to find in a random k‐uniform hypergraph. Can we then infer that H is quasi‐random? We show that the answer is negative if and only if . This resolves an open problem raised in 1991 by Chung and Graham [J AMS 4 (1991), 151–196]. While hypergraphs satisfying the property corresponding to are not necessarily quasi‐random, we manage to find a characterization of the hypergraphs satisfying this property. Somewhat surprisingly, it turns out that (essentially) there is a unique non quasi‐random hypergraph satisfying this property. The proofs combine probabilistic and algebraic arguments with results from the theory of association schemes. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Arc‐chromatic number of digraphs in which every vertex has bounded outdegree or bounded indegree

A k‐digraph is a digraph in which every vertex has outdegree at most k. A ‐digraph is a digraph in which a vertex has either outdegree at most k or indegree at most l. Motivated by function theory, we study the maximum value Φ (k) (resp. ) of the arc‐chromatic number over the k‐digraphs (resp. ‐digraphs). El‐Sahili [3] showed that . After giving a simple proof of this result, we show some better bounds. We show and where θ is the function defined by . We then study in more detail properties of Φ and . Finally, we give the exact values of and for . © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 315–332, 2006     

On Cartan's theorem for linear operators

In this paper, we describe a second main theorem of holomorphic curves in , of hyper‐order strictly less than 1, that involves a general linear operator . As an application, we derive a truncated second main theorem of degenerate holomorphic curves of hyper‐order strictly less than 1 using Nochka weights.     

A generalization of the Oberwolfach problem

Let n > 1 be an integer and let a₂,a₃,…,a_n be nonnegative integers such that . Then can be factored into ‐factors, ‐factors,…, ‐factors, plus a 1‐factor. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 151–161, 2002